A time-interleaved analog-to-digital converter (ADC) is an ADC that includes a plurality of analog-to-digital conversion circuits (AD conversion circuits), and the plurality of AD conversion circuits sequentially convert an analog input signal into a digital output signal in a time division manner and can realize high-speed operation. However, the time-interleaved ADC sometimes deteriorates in AD conversion accuracy due to a gain mismatch (difference in amplitude in each AD conversion circuit) or a skew (deviation in timing interval when each AD conversion circuit operates) between the AD conversion circuits.
As a method of correcting an error between the plurality of AD conversion circuits in the time-interleaved ADC, various methods are proposed (see, for example, Patent Documents 1 to 3, and Non-Patent Document 1). Further, there are foreground correction of performing correction processing on the error outside the normal operating time of the ADC and background correction of performing correction processing on the error during the normal operation of the ADC. In the background correction, a detection operation and a correction operation for the error can be performed without hindering a normal AD conversion operation.
FIG. 8 is a diagram illustrating a configuration example of a conventional time-interleaved ADC including a gain correction function and a skew correction function. The time-interleaved ADC includes a plurality of AD conversion circuits 801A, 801B. Assuming that the input signal into the time-interleaved ADC is cos(ωinnT+θ), an output signal y(n) from each of the AD conversion circuits 801A, 801B is expressed by the following Expression (1). When the gain of the AD conversion circuit 801A is G1 and the gain of the AD conversion circuit 801B is G2 in the Expression (1), G=(G1+G2)/2, ΔG=G1−G2. Further, ωin is an input angular frequency, ωs is an operating angular frequency (sampling angular frequency) of the time-interleaved ADC, and T is an operation cycle of the time-interleaved ADC. Further, θ is an initial phase and n is an integer equal to or larger than 0.
                              [                      Math            .                                                  ⁢            1                    ]                ⁢                                                                                                                                          y                ⁡                                  (                  n                  )                                            =                            ⁢                                                [                                      G                    +                                                                                            (                                                      -                            1                                                    )                                                n                                            ⁢                                                                        Δ                          ⁢                                                                                                          ⁢                          G                                                2                                                                              ]                                ⁢                                  cos                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        nT                                            +                      θ                                        )                                                                                                                          =                            ⁢                                                [                                      G                    +                                                                                            Δ                          ⁢                                                                                                          ⁢                          G                                                2                                            ⁢                                              cos                        ⁡                                                  (                                                                                                                    ω                                s                                                            ⁢                              nT                                                        2                                                    )                                                                                                      ]                                ⁢                                  cos                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        nT                                            +                      θ                                        )                                                                                                                          =                            ⁢                                                                    G                    ⁢                                                                                  ⁢                                          cos                      ⁡                                              (                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            nT                                                    +                          θ                                                )                                                                                                  ︸                                          input                      ⁢                                                                                          ⁢                      component                                                                      +                                                                                                    Δ                        ⁢                                                                                                  ⁢                        G                                            2                                        ⁢                    cos                    ⁢                                          {                                                                                                    (                                                                                          ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                        θ                                            }                                                                            ︸                                          image                      ⁢                                                                                          ⁢                      component                                                                                                                              (        1        )            
By multiplying the output signal y(n) by (−1)n by a multiplier 904 of a gain correction unit 802 illustrated in FIG. 9, a signal yc(n) expressed by the following Expression (2) is obtained. Multiplying the output signal y(n) by (−1)n corresponds to frequency shift of a sampling frequency fs/2 (fs=ωs/2π).
                              [                      Math            .                                                  ⁢            2                    ]                ⁢                                                                                                                                                            y                  c                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    (                                          -                      1                                        )                                    n                                ·                                  y                  ⁡                                      (                    n                    )                                                                                                                          =                            ⁢                                                cos                  ⁡                                      (                                                                                            ω                          s                                                ⁢                        nT                                            2                                        )                                                  ⁢                                  {                                                            G                      ⁢                                                                                          ⁢                                              cos                        ⁡                                                  (                                                                                                                    ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            ⁢                              nT                                                        +                            θ                                                    )                                                                                      +                                                                                            Δ                          ⁢                                                                                                          ⁢                          G                                                2                                            ⁢                                              cos                        ⁡                                                  [                                                                                                                    (                                                                                                      ω                                                                          i                                      ⁢                                                                                                                                                          ⁢                                      n                                                                                                        -                                                                                                            ω                                      s                                                                        2                                                                                                  )                                                            ⁢                              nT                                                        +                            θ                                                    ]                                                                                                      }                                                                                                        =                            ⁢                                                G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          [                                                                                                    (                                                                                          ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                        θ                                            ]                                                                      +                                                                            Δ                      ⁢                                                                                          ⁢                      G                                        2                                    ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                        θ                                            )                                                                                                                              (        2        )            
Then, by multiplying the output signal y(n) by the signal yc(n) by a multiplier 905 of the gain correction unit 802, an output expressed by the following Expression (3) is obtained.
                              [                      Math            .                                                  ⁢            3                    ]                ⁢                                                                                                            y            ⁡                          (              n              )                                ·                                    y              c                        ⁡                          (              n              )                                      =                                            {                                                G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                        θ                                            )                                                                      +                                                                            Δ                      ⁢                                                                                          ⁢                      G                                        2                                    ⁢                                      cos                    ⁡                                          [                                                                                                    (                                                                                          ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                        θ                                            ]                                                                                  }                        ×                          {                                                G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          [                                                                                                    (                                                                                          ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                        θ                                            ]                                                                      +                                                                            Δ                      ⁢                                                                                          ⁢                      G                                        2                                    ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                        θ                                            )                                                                                  }                                =                                                                      G                  2                                2                            ⁢                              {                                                      cos                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                                                  2                          ⁢                          θ                                                                    ]                                                        +                                      cos                    ⁡                                          (                                                                                                    ω                            s                                                    ⁢                          nT                                                2                                            )                                                                      }                                      +                                                                                G                    ·                    Δ                                    ⁢                                                                          ⁢                  G                                4                            ⁢                              {                                                      cos                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                              ω                                s                                                                                      )                                                    ⁢                          nT                                                +                                                  2                          ⁢                          θ                                                                    ]                                                        +                  1                                }                                      +                                                                                G                    ·                    Δ                                    ⁢                                                                          ⁢                  G                                4                            ⁢                              {                                                      cos                    ⁡                                          (                                                                        2                          ⁢                                                      ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                                                  2                          ⁢                          θ                                                                    )                                                        +                  1                                }                                      +                                          1                2                            ⁢                                                (                                                            Δ                      ⁢                                                                                          ⁢                      G                                        2                                    )                                2                            ⁢                              {                                                      cos                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                                                  2                          ⁢                          θ                                                                    ]                                                        +                                      cos                    ⁡                                          (                                                                                                    ω                            s                                                    ⁢                          nT                                                2                                            )                                                                      }                                                                        (        3        )            
In the case where the angular frequency ωin of the input signal is not ¼ of the operating angular frequency ωs of the time-interleaved ADC, terms of the trigonometric function, when averaged, can be recognized to be 0 in Expression (3). Therefore, an output from an accumulative adder 906 of the gain correction unit 802 that accumulatively adds the output y(n)·yc(n) from the multiplier 905 becomes as expressed in the following Expression (4). Using a detection value for gain correction, a multiplier 901 of the gain correction unit 802 performs gain correction.
                              [                      Math            .                                                  ⁢            4                    ]                ⁢                                                                                                                          y              ⁡                              (                n                )                                      ·                                          y                c                            ⁡                              (                n                )                                              _                =                                                                              G                  ·                  Δ                                ⁢                                                                  ⁢                G                            4                        +                                                            G                  ·                  Δ                                ⁢                                                                  ⁢                G                            4                                =                                                    G                ·                Δ                            ⁢                                                          ⁢              G                        2                                              (        4        )            
On the other hand, in the case where the angular frequency ωin of the input signal is ¼ of the operating angular frequency ωs of the time-interleaved ADC, when ωin=ωs/4 is substituted into Expression (3), the output y(n)·yc(n) from the multiplier 905 becomes as expressed in the following Expression (5).
                              [                      Math            .                                                  ⁢            5                    ]                ⁢                                                                                                            y            ⁡                          (              n              )                                ·                                    y              c                        ⁡                          (              n              )                                      =                                                            G                2                            2                        ⁢                          {                                                cos                  ⁢                                                                          ⁢                  2                  ⁢                  θ                                +                                  cos                  ⁡                                      (                                                                                            ω                          s                                                ⁢                        nT                                            2                                        )                                                              }                                +                                                                      G                  ·                  Δ                                ⁢                                                                  ⁢                G                            4                        ⁢                          {                                                cos                  ⁡                                      (                                                                  -                                                                                                            ω                              s                                                        ⁢                            nT                                                    2                                                                    +                                              2                        ⁢                        θ                                                              )                                                  +                1                            }                                +                                                                      G                  ·                  Δ                                ⁢                                                                  ⁢                G                            4                        ⁢                          {                                                cos                  ⁡                                      (                                                                                                                        ω                            s                                                    ⁢                          nT                                                2                                            +                                              2                        ⁢                        θ                                                              )                                                  +                1                            }                                +                                    1              2                        ⁢                                          (                                                      Δ                    ⁢                                                                                  ⁢                    G                                    2                                )                            2                        ⁢                          {                                                cos                  ⁡                                      (                                          2                      ⁢                      θ                                        )                                                  +                                  cos                  ⁡                                      (                                                                                            ω                          s                                                ⁢                        nT                                            2                                        )                                                              }                                                          (        5        )            
Through averaging by the accumulative adder 906, the detection value for gain correction becomes different from that in Expression (4) as illustrated in the following Expression (6).
                              [                      Math            .                                                  ⁢            6                    ]                ⁢                                                                                                                          y              ⁡                              (                n                )                                      ·                                          y                c                            ⁡                              (                n                )                                              _                =                                                            G                2                            2                        ⁢            cos            ⁢                                                  ⁢            2            ⁢            θ                    +                                                    G                ·                Δ                            ⁢                                                          ⁢              G                        2                    +                                    1              2                        ⁢                                          (                                                      Δ                    ⁢                                                                                  ⁢                    G                                    2                                )                            2                        ⁢            cos            ⁢                                                  ⁢            2            ⁢            θ                                              (        6        )            
If the multiplier 901 of the gain correction unit 802 performs gain correction using the detection value for gain correction, accurate error correction is not performed. Hence, a notch filter 903 that has a frequency characteristic as illustrated in FIG. 10 to reduce a component of ωs/4 is conventionally provided at a stage subsequent to the multiplexer 902 that selects and outputs the output relating to the AD conversion circuit 801A, 801B as illustrated in FIG. 9, so as not to perform the correction when the angular frequency ωin of the input signal is ωs/4.
Assuming that the input signal into the time-interleaved ADC is cos(ωinnT+θ), the output signal y(n) from each of the AD conversion circuits 801A, 801B containing the skew (deviation in time) is expressed by the following Expression (7). When the gain of the AD conversion circuit 801A is G1 and the gain of the AD conversion circuit 801B is G2 in the Expression (7), G=(G1+G2)/2. Further, ωin is an input angular frequency, ωs is an operating angular frequency (sampling angular frequency) of the time-interleaved ADC, and T is an operation cycle of the time-interleaved ADC. Further, Δt is a skew (deviation in time), θ is an initial phase, and n is an integer equal to or larger than 0. The following Expression (7) is expanded utilizing cos((ωinΔt)/2)≈(ωinΔt)/2, sin((ωinΔt)/2)≈1 when Δt<<1.
                              [                      Math            .                                                  ⁢            7                    ]                ⁢                                                                                                                                          y                ⁡                                  (                  n                  )                                            =                            ⁢                                                G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                        θ                                            )                                                                      ⁢                                  |                                      t                    =                                          nT                      +                                              Δ                        ⁢                                                                                                  ⁢                                                  t                          /                          2                                                                    -                                                                                                    (                                                          -                              1                                                        )                                                    n                                                ⁢                                                  (                                                      Δ                            ⁢                                                                                                                  ⁢                                                          t                              /                              2                                                                                )                                                                                                                                                                                            =                            ⁢                                                G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                2                                            )                                                        ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                +                        θ                                            )                                                                      +                                                                                                      ⁢                              G                ⁢                                                                  ⁢                                  sin                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        Δ                        ⁢                                                                                                  ⁢                        t                                            2                                        )                                                  ⁢                sin                ⁢                                  {                                                                                    (                                                                              ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                -                                                                                    ω                              s                                                        2                                                                          )                                            ⁢                      nT                                        +                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        Δ                        ⁢                                                                                                  ⁢                        t                                            2                                        +                    θ                                    }                                                                                                        ≅                            ⁢                                                                                                                  G                        ⁢                                                                                                  ⁢                                                  ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        Δ                        ⁢                                                                                                  ⁢                        t                                            2                                        ⁢                                          cos                      ⁡                                              (                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            nT                                                    +                                                                                                                    ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            ⁢                              Δ                              ⁢                                                                                                                          ⁢                              t                                                        2                                                    +                          θ                                                )                                                                                                  ︸                                          input                      ⁢                                                                                          ⁢                      component                                                                      +                                                                                                      ⁢                                                G                  ⁢                                                                          ⁢                  sin                  ⁢                                      {                                                                                            (                                                                                    ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      -                                                                                          ω                                s                                                            2                                                                                )                                                ⁢                        nT                                            +                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                2                                            +                      θ                                        }                                                                    ︸                                      image                    ⁢                                                                                  ⁢                    component                                                                                                          (        7        )            
By multiplying the output signal y(n) by (−1)n by a multiplier 807 of the skew correction unit 803, a signal yc(n) expressed by the following Expression (8) is obtained.
                              [                      Math            .                                                  ⁢            8                    ]                ⁢                                                                                                                                                            y                  c                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    (                                          -                      1                                        )                                    n                                ⁢                                  y                  ⁡                                      (                    n                    )                                                                                                                          =                            ⁢                                                cos                  ⁡                                      (                                                                                            ω                          s                                                ⁢                        nT                                            2                                        )                                                  ⁡                                  [                                                                                                                                                                                                        G                                ⁢                                                                                                                                  ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                  ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                t                                                            2                                                        ⁢                                                          cos                              ⁡                                                              (                                                                                                                                            ω                                                                              i                                        ⁢                                                                                                                                                                  ⁢                                        n                                                                                                              ⁢                                    nT                                                                    +                                                                                                                                                    ω                                                                                  i                                          ⁢                                                                                                                                                                          ⁢                                          n                                                                                                                    ⁢                                      Δ                                      ⁢                                                                                                                                                          ⁢                                      t                                                                        2                                                                    +                                  θ                                                                )                                                                                                              +                                                                                                                                                              G                          ⁢                                                                                                          ⁢                          sin                          ⁢                                                      {                                                                                                                            (                                                                                                            ω                                                                              i                                        ⁢                                                                                                                                                                  ⁢                                        n                                                                                                              -                                                                                                                  ω                                        s                                                                            2                                                                                                        )                                                                ⁢                                nT                                                            +                                                                                                                                    ω                                                                          i                                      ⁢                                                                                                                                                          ⁢                                      n                                                                                                        ⁢                                  Δ                                  ⁢                                                                                                                                          ⁢                                  t                                                                2                                                            +                              θ                                                        }                                                                                                                                ]                                                                                                        =                            ⁢                                                                                          G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    ⁢                                      cos                    ⁡                                          [                                                                                                    (                                                                                                                            ω                                  s                                                                2                                                            -                                                              ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                                                      )                                                    ⁢                          nT                                                -                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                -                        θ                                            ]                                                                      +                                                                                                      ⁢                              G                ⁢                                                                  ⁢                                  sin                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        nT                                            +                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                2                                            +                      θ                                        )                                                                                                          (        8        )            
Further, by shifting the signal yc(n) outputted from the multiplier 807 by π/2 by a phase shift circuit 808, a signal yd(n) expressed by the following Expression (9) is obtained.
                              [                      Math            .                                                  ⁢            9                    ]                ⁢                                                                                                                                                            y                  d                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                                          G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    ⁢                                      cos                    ⁡                                          [                                                                                                    (                                                                                                                            ω                                  s                                                                2                                                            -                                                              ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                                                      )                                                    ⁢                          nT                                                -                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                -                        θ                        -                                                  π                          2                                                                    ]                                                                      +                                                                                                      ⁢                              G                ⁢                                                                  ⁢                                  sin                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        nT                                            +                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                2                                            +                      θ                      -                                              π                        2                                                              )                                                                                                                          =                            ⁢                                                                                          G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    ⁢                                      sin                    ⁡                                          [                                                                                                    (                                                                                                                            ω                                  s                                                                2                                                            -                                                              ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                                                      )                                                    ⁢                          nT                                                -                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                -                        θ                                            ]                                                                      -                                                                                                      ⁢                              G                ⁢                                                                  ⁢                                  cos                  ⁡                                      (                                                                                            ω                                                      i                            ⁢                                                                                                                  ⁢                            n                                                                          ⁢                        nT                                            +                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                2                                            +                      θ                                        )                                                                                                          (        9        )            
Then, by multiplying the output signal y(n) by the signal yd(n) by a multiplier 809, an output expressed by the following Expression (10) is obtained.
                              [                      Math            .                                                  ⁢            10                    ]                ⁢                                                                                                            y            ⁡                          (              n              )                                ·                                    y              d                        ⁡                          (              n              )                                      =                                            {                                                                                          G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    ⁢                                      cos                    ⁡                                          [                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                -                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                -                        θ                                            ]                                                                      +                                  G                  ⁢                                                                          ⁢                                      sin                    ⁡                                          [                                                                                                    (                                                                                          ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                            -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                +                        θ                                            ]                                                                                  }                        ×                          {                                                                                          G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    ⁢                                      sin                    ⁡                                          [                                                                                                    (                                                                                                                            ω                                  s                                                                2                                                            -                                                              ω                                                                  i                                  ⁢                                                                                                                                          ⁢                                  n                                                                                                                      )                                                    ⁢                          nT                                                -                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                -                        θ                                            ]                                                                      -                                  G                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                                                                                    ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          nT                                                +                                                                                                            ω                                                              i                                ⁢                                                                                                                                  ⁢                                n                                                                                      ⁢                            Δ                            ⁢                                                                                                                  ⁢                            t                                                    2                                                +                        θ                                            )                                                                                  }                                =                                                    1                2                            ⁢                                                (                                                            G                      ⁢                                                                                          ⁢                                              ω                                                  i                          ⁢                                                                                                          ⁢                          n                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        2                                    )                                2                            ⁢                              {                                                      sin                    ⁡                                          (                                                                                                    ω                            s                                                    ⁢                          nT                                                2                                            )                                                        -                                      sin                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                                                                              ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                +                                                  2                          ⁢                          θ                                                                    ]                                                                      }                                      -                                                                                G                    2                                    ⁢                                      ω                                                                  i                        ⁢                                                                                                  ⁢                        n                                            ⁢                                                                                                                            ⁢                  Δ                  ⁢                                                                          ⁢                  t                                4                            ⁢                              {                                  1                  -                                      cos                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                              ω                                s                                                                                      )                                                    ⁢                          nT                                                +                                                                              ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                +                                                  2                          ⁢                          θ                                                                    ]                                                                      }                                      -                                                                                G                    2                                    ⁢                                      ω                                          i                      ⁢                                                                                          ⁢                      n                                                        ⁢                  Δ                  ⁢                                                                          ⁢                  t                                4                            ⁢                              {                                                      cos                    ⁡                                          (                                                                        2                          ⁢                                                      ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          n                          ⁢                                                                                                          ⁢                          T                                                +                                                                              ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                +                                                  2                          ⁢                          θ                                                                    )                                                        +                  1                                }                                      -                                                            G                  2                                2                            ⁢                              {                                                      sin                    ⁡                                          [                                                                                                    (                                                                                          2                                ⁢                                                                  ω                                                                      i                                    ⁢                                                                                                                                                  ⁢                                    n                                                                                                                              -                                                                                                ω                                  s                                                                2                                                                                      )                                                    ⁢                          nT                                                +                                                                              ω                                                          i                              ⁢                                                                                                                          ⁢                              n                                                                                ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                +                                                  2                          ⁢                          θ                                                                    ]                                                        +                                      sin                    ⁡                                          (                                              -                                                                                                            ω                              s                                                        ⁢                            nT                                                    2                                                                    )                                                                      }                                                                        (        10        )            
In the case where the angular frequency ωin of the input signal is not ¼ of the operating angular frequency ωs of the time-interleaved ADC, terms of the trigonometric function, when averaged, can be recognized to be 0 in Expression (10). Therefore, an output from an accumulative adder 810 that accumulatively adds the output y(n)·yd(n) from the multiplier 809 becomes as expressed in the following Expression (11). Using a detection value for skew correction, a coefficient generation unit 811 of the skew correction unit 803 generates a coefficient value using a detection value for skew correction and sets it in an adaptive filter 804 to thereby perform skew correction.
                              [                      Math            .                                                  ⁢            11                    ]                ⁢                                                                                                                          y              ⁡                              (                n                )                                      ·                                          y                d                            ⁡                              (                n                )                                              _                =                                            -                                                                                          G                      2                                        ·                                          ω                                              i                        ⁢                                                                                                  ⁢                        n                                                              ·                    Δ                                    ⁢                                                                          ⁢                  t                                4                                      -                                                                                G                    2                                    ·                                      ω                                          i                      ⁢                                                                                          ⁢                      n                                                        ·                  Δ                                ⁢                                                                  ⁢                t                            4                                =                      -                                                                                G                    2                                    ·                                      ω                                          i                      ⁢                                                                                          ⁢                      n                                                        ·                  Δ                                ⁢                                                                  ⁢                t                            2                                                          (        11        )            
On the other hand, in the case where the angular frequency ωin of the input signal is ¼ of the operating angular frequency ωs of the time-interleaved ADC, when ωin=ωs/4 is substituted into Expression (10), the output y(n)·yd(n) from the multiplier 809 becomes as expressed in the following Expression (12).
                              [                      Math            .                                                  ⁢            12                    ]                ⁢                                                                                                            y            ⁡                          (              n              )                                ·                                    y              d                        ⁡                          (              n              )                                      =                                            1              2                        ⁢                                          (                                                      G                    ⁢                                                                                  ⁢                                          ω                      s                                        ⁢                    Δ                    ⁢                                                                                  ⁢                    t                                    8                                )                            2                        ⁢                          {                                                sin                  ⁡                                      (                                                                                            ω                          s                                                ⁢                        nT                                            2                                        )                                                  -                                  sin                  ⁡                                      (                                                                                                                        ω                            s                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                4                                            +                                              2                        ⁢                        θ                                                              )                                                              }                                -                                                                      G                  2                                ⁢                                  ω                  s                                ⁢                Δ                ⁢                                                                  ⁢                t                            16                        ⁢                          {                              1                -                                  cos                  ⁡                                      (                                                                  -                                                                                                            ω                              s                                                        ⁢                            nT                                                    2                                                                    +                                                                                                    ω                            s                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                4                                            +                                              2                        ⁢                        θ                                                              )                                                              }                                -                                                                      G                  2                                ⁢                                  ω                  s                                ⁢                Δ                ⁢                                                                  ⁢                t                            16                        ⁢                          {                                                cos                  ⁡                                      (                                                                                                                        ω                            s                                                    ⁢                          nT                                                2                                            +                                                                                                    ω                            s                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                4                                            +                                              2                        ⁢                        θ                                                              )                                                  +                1                            }                                -                                                    G                2                            2                        ⁢                          {                                                sin                  ⁡                                      (                                                                                                                        ω                            s                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                          t                                                4                                            +                                              2                        ⁢                        θ                                                              )                                                  +                                  sin                  ⁡                                      (                                          -                                                                                                    ω                            s                                                    ⁢                          nT                                                2                                                              )                                                              }                                                          (        12        )            
Through averaging by the accumulative adder 810, the detection value for skew correction becomes different from that in Expression (11) as illustrated in the following Expression (13).
                              [                      Math            .                                                  ⁢            13                    ]                ⁢                                                                                                                          y              ⁡                              (                n                )                                      ·                                          y                d                            ⁡                              (                n                )                                              _                =                                            -                              1                2                                      ⁢                                          (                                                      G                    ⁢                                                                                  ⁢                                          ω                      s                                        ⁢                    Δ                    ⁢                                                                                  ⁢                    t                                    8                                )                            2                        ⁢                          sin              ⁡                              (                                                                                                    ω                        s                                            ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        4                                    +                                      2                    ⁢                    θ                                                  )                                              -                                                    G                2                            ⁢                              ω                s                            ⁢              Δ              ⁢                                                          ⁢              t                        8                    -                                                    G                2                            2                        ⁢                          sin              ⁡                              (                                                                                                    ω                        s                                            ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                        4                                    +                                      2                    ⁢                    θ                                                  )                                                                        (        13        )            
If the coefficient generation unit 811 of the skew correction unit 803 generates the coefficient value using the detection value for skew correction and sets it in the adaptive filter 804 to thereby perform skew correction, accurate error correction is not performed. Hence, a notch filter 806 that has a frequency characteristic as illustrated in FIG. 10 to reduce a component of ωs/4 is conventionally provided at a stage subsequent to the multiplexer 805 that selects and outputs the output relating to the AD conversion circuit 801A, 801B as illustrated in FIG. 8, so as not to perform the skew correction when the angular frequency ωin of the input signal is ωs/4.    [Patent Document 1] Japanese Laid-open Patent Publication No. 2013-191956    [Patent Document 2] Japanese Laid-open Patent Publication No. 2013-74308    [Patent Document 3] Japanese Laid-open Patent Publication No. 2013-31055    [Non-Patent Document 1] S. M. Jamal et al., “Calibration of Sample-Time Error in a Two-Channel Time-Interleaved Analog-to-Digital Converter,” IEEE Trans. Circuits Syst. I, vol. 51, pp. 130-139, Jan. 2004
In the above-described conventional time-interleaved ADC, in the case where the angular frequency ωin of the input signal is ¼ of the operating angular frequency ωs of the time-interleaved ADC, the gain correction and the skew correction is not accurately performed. This is because when ωin*≠ωs/4, a component INPUT of the input signal and a component (image signal component) IMAGE of the error signal in the digital output signal become different frequency components as illustrated in FIG. 11A, whereas when ωin=ωs/4 the component INPUT of the input signal and the component (image signal component) IMAGE of the error signal in the digital output signal become the same frequency components as illustrated in FIG. 11B and is not distinguished from each other.